Mixed Boundary Value Problems for Stationary Magnetohydrodynamic Equations of a Viscous Heat-Conducting Fluid

We consider the boundary value problem for stationary magnetohydrodynamic equations of electrically and heat conducting fluid under inhomogeneous mixed boundary conditions for electromagnetic field and temperature and Dirichlet condition for the velocity. The problem describes the thermoelectromagnetic flow of a viscous fluid in 3D bounded domain with the boundary consisting of several parts with different thermo- and electrophysical properties. The global solvability of the boundary value problem is proved and the apriori estimates of the solution are derived. The sufficient conditions on the data are established which provide a local uniqueness of the solution.     

Estimates of Solutions and Asymptotic Symmetry for Parabolic Equations on Bounded Domains

We consider fully nonlinear parabolic equations on bounded domains under Dirichlet boundary conditions. Assuming that the equation and the domain satisfy certain symmetry conditions, we prove that each bounded positive solution of the Dirichlet problem is asymptotically symmetric. Compared with previous results of this type, we do not assume certain crucial hypotheses, such as uniform (with respect to time) positivity of the solution or regularity of the nonlinearity in time. Our method is based on estimates of solutions of linear parabolic problems, in particular on a theorem on asymptotic positivity of such solutions.     

Existence and Uniqueness Theorems for the Two-Dimensional Ericksen–Leslie System

In this paper we study the two dimensional Ericksen–Leslie equations for the nematodynamics of liquid crystals if the moment of inertia of the molecules does not vanish. We prove short time existence and uniqueness of strong solutions for the initial value problem in two situations: the space-periodic problem and the case of a bounded domain with spatial Dirichlet boundary conditions on the Eulerian velocity and the cross product of the director field with its time derivative. We also show that the speed of propagation of the director field is finite and give an upper bound for it.     

Résumé and remarks on the open boundary condition minisymposium

The incompressible Navier-Stokes equations—and their thermal convection and stratified flow analogue, the Boussinesq equations—possess solutions in bounded domains only when appropriate/legitimate boundary conditions (BCs) are appended at all points on the domain boundary. When the boundary—or, more commonly, a portion of it—is not endowed with a Dirichlet BC, we are faced with selecting what are called open boundary conditions (OBCs), because the fluid may presumably enter or leave the domain through such boundaries. The two minisymposia on OBCs that are summarized in this paper had the objective of finding the best OBCs for a small subset of two-dimensional test problems. This objective, which of course is not really well-defined, was not met (we believe), but the contributions obtained probably raised many more questions/issues than were resolved—notable among them being the advent of a new class of OBCs that we call FBCs (fuzzy boundary conditions).     

Stability of Semiconductor States with Insulating and Contact Boundary Conditions

We prove the existence of global smooth solutions near a given steady state of the hydrodynamic model of the semiconductors in a bounded domain with physical boundary conditions. The steady state and the doping profile are permitted to be of large variation but the initial velocity must be small. Two cases are considered. In the first one the problem is three-dimensional, the boundary conditions are insulating and the steady state velocity vanishes. In the second one, the problem is one-dimensional, the boundary is of contact type and the steady state velocity does not vanish.     

On the Nonexistence of Some Special Eigenfunctions for the Dirichlet Laplacian and the Lamé System

Let Ω be a bounded Lipschitz domain in ℝ ⁿ with n ≥ 3. We prove that the Dirichlet Laplacian does not admit any eigenfunction of the form u(x) =ϕ(x′)+ψ(x _n) with x′=(x1, ..., x _n−1). The result is sharp since there are 2-d polygonal domains in which this kind of eigenfunctions does exist. These special eigenfunctions for the Dirichlet Laplacian are related to the existence of uniaxial eigenvibrations for the Lamé system with Dirichlet boundary conditions. Thus, as a corollary of this result, we deduce that there is no bounded Lipschitz domain in 3-d for which the Lamé system with Dirichlet boundary conditions admits uniaxial eigenvibrations. This revised version was published online in August 2006 with corrections to the Cover Date.     

Global Boundedness of the Gradient for a Class of Nonlinear Elliptic Systems

Gradient boundedness up to the boundary for solutions to Dirichlet and Neumann problems for elliptic systems with Uhlenbeck type structure is established. Nonlinearities of possibly non-polynomial type are allowed, and minimal regularity on the data and on the boundary of the domain is assumed. The case of arbitrary bounded convex domains is also included.     

Stability analysis of numerical interface conditions in fluid–structure thermal analysis

This paper analyses the numerical stability of coupling procedures in modelling the thermal diffusion in a solid and a fluid with continuity of temperature and heat flux at the interface. A simple one-dimensional model is employed with uniform material properties and grid density in each domain. A number of different explicit and implicit algorithms are considered for both the interior equations and the boundary conditions. The analysis shows that in general these are stable provided that Dirichlet boundary conditions are imposed on the fluid and Neumann boundary conditions are imposed on the solid; in each case the imposed values are obtained from the other domains. © 1997 John Wiley & Sons, Ltd.     

Regularity of Flow of Anisotropic Fluid

We study the steady flow of an anisotropic generalised Newtonian fluid under Dirichlet boundary conditions in a bounded domain . The fluid is characterised by a nonlinear dependence of the stress tensor on the symmetric gradient of the velocity vector field. We prove the existence of a C ^1,α-solution of this problem under certain assumptions on the growth of the elliptic term. The result is global: we prove the regularity up to the boundary of the domain Ω. This research was supported by the grant GACR 201/03/0934; partially also by the grants MSM 0021620839 and GAUK 262/2002/B-MAT/MFF.     

Existence of Weak Solutions for the Motion of Rigid Bodies in a Viscous Fluid

. We study the evolution of a finite number of rigid bodies within a viscous incompressible fluid in a bounded domain of with Dirichlet boundary conditions. By introducing an appropriate weak formulation for the complete problem, we prove existence of solutions for initial velocities in . In the absence of collisions, solutions exist for all time in dimension 2, whereas in dimension 3 the lifespan of solutions is infinite only for small enough data. (Accepted June 10, 1998)     

A ghost fluid Lattice Boltzmann method for complex geometries

A ghost fluid Lattice Boltzmann method (GF‐LBM) is developed in this study to represent complex boundaries in Lattice Boltzmann simulations of fluid flows. Velocity and density values at the ghost points are extrapolated from the fluid interior and domain boundary via obtaining image points along the boundary normal inside the fluid domain. A general bilinear interpolation algorithm is used to obtain values at image points which are then extrapolated to ghost nodes thus satisfying hydrodynamic boundary conditions. The method ensures no‐penetration and no‐slip conditions at the boundaries. Equilibrium distribution functions at the ghost points are computed using the extrapolated values of the hydrodynamic variables, while non‐equilibrium distribution functions are extrapolated from the interior nodes. The method developed is general, and is capable of prescribing Dirichlet as well as Neumann boundary conditions for pressure and velocity. Consistency and second‐order accuracy of the method are established by running three test problems including cylindrical Couette flow, flow between eccentric rotating cylinders and flow over a cylinder in a confined channel. Copyright © 2011 John Wiley & Sons, Ltd.     

Dirichlet and Neumann boundary conditions for the pressure poisson equation of incompressible flow

In a recent paper Gresho and Sani showed that Dirichlet and Neumann boundary conditions for the pressure Poisson equation give the same solution. The purpose of this paper is to confirm this (for one case at least) by numerically solving the pressure equation with Dirichlet and Neumann boundary conditions for the inviscid stagnation point flow problem. The Dirichlet boundary condition is obtained by integrating the tangential component of the momentum equation along the boundary. The Neumann boundary condition is obtained by applying the normal component of the momentum equation at the boundary. In this work solutions for the Neumann problem exist only if a compatibility condition is satisfied. A consistent finite difference procedure which satisfies this condition on non-staggered grids is used for the solution of the pressure equation with Neumann conditions. Two test cases are computed. In the first case the velocity field is given from the analytical solution and the pressure is recovered from the solution of the associated Poisson equation. The computed results are identical for both Dirichlet and Neumann boundary conditions. However, the Dirichlet problem converges faster than the Neumann case. In the second test case the velocity field is computed from the momentum equations, which are solved iteratively with the pressure Poisson equation. In this case the Neumann problem converges faster than the Dirichlet problem.     

Vortex dynamics on a domain with holes

We explore the relationship between the hydrodynamic Green’s function and the Dirichlet Green’s function in a bounded domain with holes. This relationship is expressed using the harmonic measures on the domain, following the work of Flucher and Gustafsson (Vortex motion in two-dimensional hydrodynamics, energy renormalization and stability of vortex pairs, TRITA preprint series, 1997). The explicit form of this relation expresses velocity in terms of vorticity in a way which turns out to be very convenient, especially for analysis. We explain the advantages and describe some applications.     

Vortex particle‐in‐cell method for three‐dimensional viscous unbounded flow computations

A new vortex particle‐in‐cell (PIC) method is developed for the computation of three‐dimensional unsteady, incompressible viscous flow in an unbounded domain. The method combines the advantages of the Lagrangian particle methods for convection and the use of an Eulerian grid to compute the diffusion and vortex stretching. The velocity boundary conditions used in the method are of Dirichlet‐type, and can be calculated using the vorticity field on the grid by the Biot–Savart equation. The present results for the propagation speed of the single vortex ring are in good agreement with the Saffman's model. The applications of the method to the head‐on and head‐off collisions of the two vortex rings show good agreement with the experimental and numerical literature. Copyright © 2000 John Wiley & Sons, Ltd.     

Efficient quadrature for a boundary element method to compute three-dimensional Stokes flow

A collocation-type boundary element method based on bilinear B-splines is used for the numerical solution of the Stokes Dirichlet problem in bounded domains D ? R³. The computation of the influence matrix requires the numerical evaluation of weakly singular integrals on the domain boundary if the usual double-layer potential ansatz is chosen. Here mostly standard methods with disjoint grids for collocation and integration are used. We develop a special integration scheme based on triangular co-ordinates near the singularity and show its efficiency compared with the method mentioned above.     

Concentration of Solutions for Some Singularly Perturbed Mixed Problems: Existence Results

In this paper,we study the asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions. We prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero.     

On the Large Time Behavior of Solutions of Hamilton–Jacobi Equations Associated with Nonlinear Boundary Conditions

In this article, we study the large time behavior of solutions of first-order Hamilton–Jacobi Equations set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy–Neumann problems by using two fairly different methods: the first one relies only on partial differential equations methods, which provides results even when the Hamiltonians are not convex, and the second one is an optimal control/dynamical system approach, named the “weak KAM approach”, which requires the convexity of Hamiltonians and gives formulas for asymptotic solutions based on Aubry–Mather sets.     

Potential flow around obstacles using the scaled boundary finite‐element method

The scaled boundary finite‐element method is a novel semi‐analytical technique, combining the advantages of the finite element and the boundary element methods with unique properties of its own. The method works by weakening the governing differential equations in one co‐ordinate direction through the introduction of shape functions, then solving the weakened equations analytically in the other (radial) co‐ordinate direction. These co‐ordinate directions are defined by the geometry of the domain and a scaling centre. The method can be employed for both bounded and unbounded domains. This paper applies the method to problems of potential flow around streamlined and bluff obstacles in an infinite domain. The method is derived using a weighted residual approach and extended to include the necessary velocity boundary conditions at infinity. The ability of the method to model unbounded problems is demonstrated, together with its ability to model singular points in the near field in the case of bluff obstacles. Flow fields around circular and square cylinders are computed, graphically illustrating the accuracy of the technique, and two further practical examples are also presented. Comparisons are made with boundary element and finite difference solutions. Copyright © 2003 John Wiley & Sons, Ltd.     

On the stability and dissipation of wall boundary conditions for compressible flows

Characteristic formulations for boundary conditions have demonstrated their effectiveness to handle inlets and outlets, especially to avoid acoustic wave reflections. At walls, however, most authors use simple Dirichlet or Neumann boundary conditions, where the normal velocity (or pressure gradient) is set to zero. This paper demonstrates that there are significant differences between characteristic and Dirichlet methods at a wall and that simulations are more stable when using walls modelled with a characteristic wave decomposition. The derivation of characteristic methods yields an additional boundary term in the continuity equation, which explains their increased stability. This term also allows to handle the two acoustic waves going towards and away from the wall in a consistent manner. Those observations are confirmed by stability matrix analysis and one‐ and two‐dimensional simulations of acoustic modes in cavities. Copyright © 2009 John Wiley & Sons, Ltd.     

基于FTM方法的二维Kelvin-Helmholtz不稳定性数值模拟

使用界面跟踪法FTM（Front Tracking Method）对二维不混溶、不可压缩流体的K-H（Kelvin-Helmholtz）不稳定性进行数值模拟。研究表明,速度梯度层越厚,界面在水平分量中移动越快,卷起越少;初始水平速度差越大,界面卷起越多,内扰动增长速度越快,K-H不稳定性的特征形式更加明显;此外,在Neumann边界条件（即无滑移边界条件）下界面的扰动发展得比Dirichlet边界条件（即对称边界条件）下的扰动快。由于Dirichlet边界中的边界层,在开始时刻涡量扩展到两侧,影响了K-H不稳定性的生长速率;而在Neumann边界条件下涡量由于初始水平速度差,在界面中心聚集。最后,研究了不同边界条件下各种理查德森数对K-H不稳定性的影响。